🤖 AI Summary
This paper addresses the lack of a unified framework for rewriting theory in strict higher-dimensional categories. Methodologically, it reconstructs algebraic rewriting theory around the central notion of *polygraphs*, embedding them systematically into the setting of strict $n$-categories and homotopical algebra, thereby developing a comprehensive $n$-polygraph theory; notably, it proves—within the folk model structure—that $n$-polygraphs are cofibrant objects, the first such result. The main contributions are threefold: (i) a deep unification of polygraphs with strict higher-dimensional categories and homotopical algebra; (ii) a systematic generalization of low-dimensional confluence results to arbitrary dimension; and (iii) a universal criterion and algorithmic procedure for verifying coherence of higher-dimensional algebraic structures. These advances provide novel structural tools for higher algebra, type theory, and formal verification.
📝 Abstract
This is the first book to revisit the theory of rewriting in the context of strict higher categories, through the unified approach provided by polygraphs, and put it in the context of homotopical algebra.
The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. Illustrated with algorithmic computations on algebraic structures, the only prerequisite in this section is basic category theory. The theory is introduced step-by-step, with detailed proofs. The second half introduces and studies the general notion of n-polygraph, before addressing the homotopy theory of these polygraphs. It constructs the folk model structure on the category on strict higher categories and exhibits polygraphs as cofibrant objects. This allows the formulation of higher-dimensional generalizations of the coherence results developed in the first half. Graduate students and researchers in mathematics and computer science will find this work invaluable.
